Graded Skew Clifford Algebras that are Twists of Graded Clifford Algebras

TL;DR

This paper investigates the structure of regular graded skew Clifford algebras that are twists of graded Clifford algebras, showing how certain subalgebras relate to polynomial rings and providing examples of when this relationship fails.

Contribution

It establishes conditions under which subalgebras of twisted regular graded skew Clifford algebras are skew polynomial rings, and highlights limitations through counterexamples.

Findings

Subalgebra generated by normalizing sequence is a skew polynomial ring when A is a twist of B.

The subalgebra is a twist of a polynomial ring by an automorphism.

Counterexample shows the property can fail without the twist condition.

Abstract

We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$.